I'm currently studying Liouville's theorem in differential algebra from this paper. I'm stuck on the proof of the propositions 4.2.
In the chapter 4, this "logarithmic derivative identity" is introduced: $$\frac{\left(\prod_{j=1}^nt_j^{m_j}\right)^\prime}{\prod_{j=1}^nt_j^{m_j}}=\sum_{j=1}^nm_j\frac{t_j^\prime}{t_j}$$
According to the proposition 4.2, if the equation $$\alpha = \sum_{j=1}^mc_j\frac{\left(q_j(\ell)\right)^\prime}{q_j(\ell)}+\left(r(\ell)\right)^\prime$$ holds, then the $q_j(t)$ are either non-constant monic irreducible polynomials in $K[\ell]$ or elements of $K$.
$L=K(\ell)$ is a differential field extension, where $\ell$ is transcendental over $K$; $\alpha \in K$; $c_1,\ldots,c_m\in K_C$ are constants/elements of the kernel of the derivation; and $r(\ell), q_j(\ell)\in L$.
The proof says that the proposition follows from the logarithmic derivative identity, but I just can't see, how.
I did try writing the equation as $$\alpha= \frac{\left(\prod_{j=1}^m\left(q_j(\ell)\right)^{c_j}\right)^\prime}{\prod_{j=1}^m\left(q_j(\ell)\right)^{c_j}} +\left(r(\ell)\right)^\prime$$ and substituting $q_j(\ell)=k_j\prod_{i=1}^{n_j}\left(q_{ji}(\ell)\right)^{n_{ji}}$, but that didn't seem to lead anywhere.
As explained in the paper, we can first write each $q_j(\ell)$ as a quotient of two polynomials in $\ell$, which we can factorise into monic irreducible polynomials. Thus, we can write $$q_j(\ell)=k_j\prod_{i=1}^{n_j}\left(q_{ji}\left(\ell\right)\right)^{n_{ji}}=k_j\cdot\hat{q}_j(\ell),$$ where $k_j\in K$, $q_{ji}(\ell)\in K[\ell]$, and $n_{ji}$ are integers not necessarily non-negative. Then, $$\frac{(q_j(\ell))^\prime}{q_j(\ell)}=\frac{\left(k_j\hat{q}_j(\ell)\right)^\prime}{k_j\hat{q}_j(\ell)}=\frac{k_j^\prime\hat{q}_j(\ell)+k_j\hat{q}_j^\prime(\ell)}{k_j\hat{q}_j(\ell)}=\frac{k_j^\prime}{k_j}+\frac{\hat{q}_j^\prime(\ell)}{\hat{q}_j(\ell)}=\frac{k_j^\prime}{k_j}+\frac{\left(\prod_{i=1}^{n_j}\left(q_{ji}\left(\ell\right)\right)^{n_{ji}}\right)^\prime}{\prod_{i=1}^{n_j}\left(q_{ji}\left(\ell\right)\right)^{n_{ji}}}.$$ We can apply the logarithmic derivative identity on the last term: $$\frac{(q_j(\ell))^\prime}{q_j(\ell)}=\frac{k_j^\prime}{k_j}+\sum_{i=1}^{n_j}n_{ji}\frac{\left(q_{ji}(\ell)\right)^\prime}{q_{ji}(\ell)}.$$ It is clear that when we multiply this by $c_j$ and sum over $j$, the sum is in the form we wanted.